152 CES TRANSACTIONS ON ELECTRICAL MACHINES AND SYSTEMS, VOL. 2, NO. 1, MARCH 2018 Abstract—In this paper, a fast design method is developed based on a combination of analytical and finite element (FE) methods for variable flux reluctance machines (VFRMs). Firstly, the feasibility of using analytical method in optimization under unsaturated condition is confirmed. Then, by applying the FE method, the influence of magnetic saturation is considered. Compared with the unsaturated case, the optimal split ratio for magnetically saturated case is increased by 1~1.2 times, the optimal rotor pole arc ratio varies within 0.33~0.44, and the stator pole arc ratio remains the same. Based on this, the optimal structural parameters can be initially set by analytical method and then refined by the FE method. Due to the fast speed of analytical method, less variable counts and narrowed variation ranges, the proposed method is significantly faster than the conventional pure FE based global optimization. Finally, the proposed method is used for optimizing the 6-stator-slots VFRMs having different numbers of rotor poles. The 6-stator-slot/7-rotor-pole (6s/7r) VFRM is found to have the highest torque density. It is prototyped and tested to verify the analyses. Index Terms—Analytical method, optimal design, torque density, variable flux reluctance machine. NOMENCLATURE As Total stator slot area F Magnetomotive force (MMF) Fa, Ff MMFs of armature and field currents Fs Modulated MMF Fsa, Fsf Modulated MMFs of armature and field currents g0 Airgap length hs, hr Yoke thickness of stator and rotor Ia, Ib, Ic Currents of phases A, B and C If Current of field winding Irms, Idc RMS values of armature and field currents kT Torque coefficient Lstk Machine stack length Pcu Total copper loss of VFRM Pcua, Pcuf Copper losses of armature and field currents Rsi Stator inner radius This article was submitted for review on 26, January, 2018. L.R. Huang and Z.Q. Zhu (corresponding author) are with the Electrical Machines and Drives Group, University of Sheffield, Sheffield S1 3JD, U.K. (e-mail: [email protected]; z.q.zhu@sheffield. ac.uk). J.H. Feng, S.Y. Guo, J.X. Shi and W.Q. Chu are with the CRRC Zhuzhou Institute Co. Ltd, Shidai Road, Shifeng District, Zhuzhou, Hunan, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected];). Rso Stator outer radius Te Electromagnetic torque Ts, Tr, Tc Synchronous/reluctance/cogging torques Wa, Wb, Wc Winding functions of phases A, B and C Wf Winding function of field winding Wt Stator tooth width βs, βr Stator and rotor slot opening ratios θ Mechanical angle θs Stator slot pitch λ Split ratio Λr1 Magnitude of fundamental rotor radial permenace component Λs, Λr Stator and rotor permeance functions obtained by single-side saliency model μ0 Vacuum permeability I. INTRODUCTION UE to the increasing concerns on the price of rare-earth magnet material and the risk of demagnetization in permanent magnet (PM) machines [1-3], many magnetless machines, including induction machines (IMs) [4], rotor-wound-field synchronous machines (RWFSMs) [5-6], switched reluctance machines (SRMs) [7-8], synchronous reluctance machine (SynRMs) [9-10], vernier reluctance machines (VRM) [11], stator-wound-field flux switching machines (SWFFSMs) [12-13], and variable flux reluctance machines (VFRMs) [14-18] have been extensively investigated. VFRMs are developed in [14] and [15]. Fig. 1 shows the configurations of two typical VFRMs, i.e. 6-stator-slot/4-rotor-pole (6s/4r) and 6s/5r VFRMs. They have doubly-salient structure, which are similar to that of switched reluctance machine (SRM), and two sets of concentrated windings, i.e., AC armature and DC field windings. Apart from the advantages inherited from SRMs, such as robust rotor and compact windings, VFRMs show significantly smaller torque ripple and acoustic noise [16], and more flexible rotor pole number selection [17-18] than the SRMs. Moreover, the stator-located winding structure avoids the requirement of slip-rings/brushes and the heat can be easily dissipated from the stator. All these merits extend the application of VFRMs. In order to obtain the highest torque density, design methods of VFRMs are important. In [19], the torque density of a 48s/40r VFRM is proved to be closely related to the rotor pole arc and rotor tooth height. In [20], four 6-stator-pole VFRMs Fast Design Method of Variable Flux Reluctance Machines L.R. Huang, J.H. Feng, S.Y. Guo, J.X. Shi, W.Q. Chu, and Z.Q. Zhu, Fellow, IEEE D
Transcript
152 CES TRANSACTIONS ON ELECTRICAL MACHINES AND SYSTEMS, VOL. 2, NO. 1, MARCH 2018
Abstract—In this paper, a fast design method is developed
based on a combination of analytical and finite element (FE) methods for variable flux reluctance machines (VFRMs). Firstly, the feasibility of using analytical method in optimization under unsaturated condition is confirmed. Then, by applying the FE method, the influence of magnetic saturation is considered. Compared with the unsaturated case, the optimal split ratio for magnetically saturated case is increased by 1~1.2 times, the optimal rotor pole arc ratio varies within 0.33~0.44, and the stator pole arc ratio remains the same. Based on this, the optimal structural parameters can be initially set by analytical method and then refined by the FE method. Due to the fast speed of analytical method, less variable counts and narrowed variation ranges, the proposed method is significantly faster than the conventional pure FE based global optimization. Finally, the proposed method is used for optimizing the 6-stator-slots VFRMs having different numbers of rotor poles. The 6-stator-slot/7-rotor-pole (6s/7r) VFRM is found to have the highest torque density. It is prototyped and tested to verify the analyses.
NOMENCLATURE As Total stator slot area F Magnetomotive force (MMF) Fa, Ff MMFs of armature and field currents Fs Modulated MMF Fsa, Fsf Modulated MMFs of armature and field
currents g0 Airgap length hs, hr Yoke thickness of stator and rotor Ia, Ib, Ic Currents of phases A, B and C If Current of field winding Irms, Idc RMS values of armature and field currents kT Torque coefficient Lstk Machine stack length Pcu Total copper loss of VFRM Pcua, Pcuf Copper losses of armature and field currents Rsi Stator inner radius
This article was submitted for review on 26, January, 2018. L.R. Huang and Z.Q. Zhu (corresponding author) are with the Electrical
Machines and Drives Group, University of Sheffield, Sheffield S1 3JD, U.K. (e-mail: [email protected]; z.q.zhu@sheffield. ac.uk).
Te Electromagnetic torque Ts, Tr, Tc Synchronous/reluctance/cogging torques Wa, Wb, Wc Winding functions of phases A, B and C Wf Winding function of field winding Wt Stator tooth width βs, βr Stator and rotor slot opening ratios θ Mechanical angle θs Stator slot pitch λ Split ratio Λr1 Magnitude of fundamental rotor radial
permenace component Λs, Λr Stator and rotor permeance functions obtained
by single-side saliency model μ0 Vacuum permeability
I. INTRODUCTION UE to the increasing concerns on the price of rare-earth magnet material and the risk of demagnetization in
permanent magnet (PM) machines [1-3], many magnetless machines, including induction machines (IMs) [4], rotor-wound-field synchronous machines (RWFSMs) [5-6], switched reluctance machines (SRMs) [7-8], synchronous reluctance machine (SynRMs) [9-10], vernier reluctance machines (VRM) [11], stator-wound-field flux switching machines (SWFFSMs) [12-13], and variable flux reluctance machines (VFRMs) [14-18] have been extensively investigated.
VFRMs are developed in [14] and [15]. Fig. 1 shows the configurations of two typical VFRMs, i.e. 6-stator-slot/4-rotor-pole (6s/4r) and 6s/5r VFRMs. They have doubly-salient structure, which are similar to that of switched reluctance machine (SRM), and two sets of concentrated windings, i.e., AC armature and DC field windings. Apart from the advantages inherited from SRMs, such as robust rotor and compact windings, VFRMs show significantly smaller torque ripple and acoustic noise [16], and more flexible rotor pole number selection [17-18] than the SRMs. Moreover, the stator-located winding structure avoids the requirement of slip-rings/brushes and the heat can be easily dissipated from the stator. All these merits extend the application of VFRMs.
In order to obtain the highest torque density, design methods of VFRMs are important. In [19], the torque density of a 48s/40r VFRM is proved to be closely related to the rotor pole arc and rotor tooth height. In [20], four 6-stator-pole VFRMs
Fast Design Method of Variable Flux Reluctance Machines
HUANG et al: FAST DESIGN METHOD OF VARIABLE FLUX RELUCTANCE MACHINES 153
with 4, 5, 7 and 8-rotor poles are globally optimized. It is concluded that the maximum torque is achieved when the optimal rotor pole arc to pole pitch ratio is around 1/3 and the stator pole arc is equal or slightly smaller than rotor pole arc. Then, a weighted evaluation function is introduced in [21] to take the torque density, torque ripple, cogging torque, power factor, field winding voltage fluctuation and copper consumption into account during the optimization of stator wound field synchronous machine. However, in all these existing works, the optimization purely relies on the finite element (FE) based global optimization method, which is known to be time consuming.
In this paper, a fast optimization method is developed by the synergy of analytical and FE methods. In Section II, the feasibility of using analytical method in machine optimization under magnetically unsaturated condition is verified. In Section III, the influence of magnetic saturation on the optimal structural parameters is investigated by FE analyses. Based on this, a fast optimization method is developed in Section IV. It is then employed to optimize the 6-stator-pole VFRMs having different numbers of rotor poles to verify its capability. Finally, the experimental validation on a 6s/7r VFRM is presented in Section V.
II. ANALYTICAL OPTIMIZATION METHOD In this section, the optimization by analytical method is
discussed. Since the analytical method can only be applicable to the linear case (the permeability of cores is set to infinity), the influence of magnetic saturation will be investigated later in Section III.
A. Analytical torque calculation model Based on the Lorentz force law, the instantaneous torque
expression of VFRMs is given by (1). The detailed derivation procedure can be found in [22].
( ) ( ) ( ) ( )2
0, , ,e si stk s r sT t R L F t t dF t
πθ θ θ= − Λ∫ (1)
where Rsi is the radius of stator inner surface; Lstk is the machine stack length; θ is the mechanical angle in the stator reference frame; Λr is the rotor radial permeance function obtained by salient rotor and slotless stator model; and Fs is the “Modulated MMF” defined by
( ) ( ) ( )0 0, ,s sF t F t gθ θ θ µ= Λ (2) where g0 is the airgap length; μ0 is the vacuum permeability; Λs
(θ, t) is the stator radial permeance obtained by smooth rotor and slotted stator model; and F(θ, t) is the MMF generated by the armature and field windings, i.e.
( ) ( ) ( ), , ,a fF t F t F tθ θ θ= + (3) where Fa (θ, t) and Ff (θ, t) are the MMF functions [23] of armature winding and field winding, respectively. They can be deduced by the product of corresponding winding functions and excitations, i.e.
( ) ( ) ( ) ( ) ( ) ( )( ) ( )
( , t)
( , t)a a a b b c c
f f f
F W I t W I t W I t
F W I t
θ θ θ θ
θ θ
= + +
=
(4)
where Wa (θ), Wb (θ), Wc (θ) and Wf (θ), Ia (t), Ib (t), Ic (t) and If (t) are the winding functions and currents of phase A, phase B, phase C and field windings, respectively.
Regarding the stator and rotor radial permeance, both of
(a)
(b)
Fig. 1. Cross sections and winding configurations of the 6s/4r and 6s/5r VFRMs. (a) 6s/4r VFRM. (b) 6s/5r VFRM
Fig. 2. Distributions of analytically and FE predicted stator radial permeances for different stator slot opening ratios (Rsi=22.5mm, g0=0.5mm, θs=60deg.).
TABLE I MAIN SPECIFICATIONS OF 6S/4R AND 6S/5R VFRMS
Parameter Unit VFRM 6s/4r 6s/5r
Stator outer diameter mm 90 Airgap length mm 0.5
Total copper loss W 30 Turns per coil (AC/DC) - 144/144
Split ratio - 0.5 0.52 Stator pole arc deg. 27 24 Rotor pole arc deg. 34.6 26
Fig. 3. Torque waveforms of optimized 6s/4r and 6s/5r VFRMs predicted by analytical and FEA methods (Pcu=30W).
A1A
2
B1
B2
C1
C2
DC
1
DC2DC3
DC
4
DC5 DC6
StatorRotor
Field windingArmature winding
A1A
2
B1
B2
C1
C2
DC
1
DC2DC3
DC
4
DC5 DC6
StatorRotor
Field windingArmature winding
0
0.5
1
1.5
2
2.5
3
0 10 20 30 40 50 60
Stat
or r
adia
l per
mea
nce
(10-3
)
Mechnical angle (deg.)
FEA Analytical
0.78sβ =
0.67sβ =
0.44sβ =
0.56sβ =
0
0.2
0.4
0.6
0.8
1
1.2
0 60 120 180 240 300 360
Out
put t
orqu
e (N
m)
Rotor position (Elec. deg.)
FEA Analytical
6s/4r
6s/5r
154 CES TRANSACTIONS ON ELECTRICAL MACHINES AND SYSTEMS, VOL. 2, NO. 1, MARCH 2018
them can be calculated by the analytical model for single-sided saliency motor [24]. Taking the stator radial permeance for example, its distribution under one stator slot pitch is
( ) ( )
( )( ) ( )
( ) ( ) [ ]
[ ]
0 0
sin sin 2 2, 0,
2 sin 4 cos 2 4
0 , ,
s
s ssis s
s s s s
s s s
g g
Rg
θ µ θ
θ β θ θπ θ β θβ θ θ β θθ
θ β θ θ
Λ = + −
∈ −= ∈
(5)
where βs is the stator slot opening ratio and θs is the slot pitch. A comparison between analytically and FE predicted stator
radial permeance distributions over one stator slot pitch for different stator slot opening ratios is presented in Fig. 2. Good accuracy is found for the analytical method.
Similarly, the rotor radial permeance function can also be deduced but is not presented here for simplicity.
After obtaining the permeance and MMF distributions, the instantaneous torque of VFRMs can be predicted analytically. For verification, the 6s/4r and 6s/5r VFRMs are chosen as examples. Their main specifications are listed in Table I. Fig. 3 compares the analytically and FE predicted torque waveforms. Good agreement is observed, indicating the accuracy of using analytical method in torque estimation under magnetic unsaturation condition B. Optimal AC/DC ratio for maximum torque
Further, by substituting (2) and (3) into (1), the torque equation can be divided into three components: synchronous torque Ts, reluctance torque Tr and cogging torque Tc, as shown in (6).
(6)
where Fsa and Fsf are modulated MMFs of armature and field currents, respectively.
Based on the harmonic analysis, the torque principle of VFRMs is comprehensively illustrated with the concept of magnetic gearing effect in [22]. It is found that the average torque of VFRMs is mainly generated by synchronous torque and can be concisely expressed by:
_ _ 1e avg s avg T si stk r r rms dcT T k R L N I I≈ = Λ (7) where kT is a coefficient determined by the stator radial permeance and winding functions; Λr1 is the magnitude of the 1st rotor radial permeance harmonic; Irms and Idc are the rms values of armature and field currents, respectively.
Assuming the total copper loss (the sum of armature copper loss Pcua and field copper loss Pcuf) of VFRM, Pcu, is a constant, i.e.
2 2cu cua cuf rms a dc fP P P I R I R= + = + (8)
where Ra and Rf are the total resistances of armature and field windings, respectively.
Then, the torque equation (7) can be rewritten as:
( )2_ 1e avg e si stk r r rms cu rms a fT k R L N I P I R R= Λ − (9)
In order to maximize the average torque, dTe_avg/dIrms=0:
Fig. 4. Relationship between optimization parameters and torque equation variables.
(a)
(b)
(c)
(d)
Fig. 5. Variation of average torques against stator and rotor pole arcs for 6s/4r and 6s/5r VFRMs (Split ratio=0.5). (a) Analytical results of 6s/4r VFRM. (b) FE results of 6s/4r VFRM. (c) Analytical results of 6s/5r VFRM. (d) FE results of 6s/5r VFRM.
Fig. 6. Variation of 1st rotor radial permeance harmonic and average torques of 6s/4r and 6s/5r VFRMs against rotor pole arc ratio under linear condition.
TABLE II CONSTRAINTS FOR OPTIMIZATION
Parameter Symbol Unit Value Stator outer radius Rso mm 45
Stator/rotor yoke thickness hs / hr mm ≥5 Airgap length g0 mm 0.5
Total copper loss Pcu W 30 Turns per coil (AC/DC) na/nf - 144/144
Stack length Lstk mm 25
TABLE III OPTIMAL STRUCTURAL PARAMETERS FOR 6S/4R AND 6S/5R VFRMS UNDER
HUANG et al: FAST DESIGN METHOD OF VARIABLE FLUX RELUCTANCE MACHINES 155
( )( )
1/22
_ 1 1/22 2
2
0
2 2
cu rms a
e avg rms e si stk r r r
rms a cu rms a
cu rms a cua
cuf cu cua cua
P I RdT dI k R L N N
I R P I R
P I R PP P P P
−
− − = Λ = −
⇒ = =⇒ = − =
(10)
Therefore, the largest average torque/copper loss can be obtained when the copper losses of field and armature windings are the same, i.e. Pcua=Pcuf. Further, if the field and armature windings share the same slot area and turns number, the optimal AC/DC ratio is Irms/Idc=1. This conclusion is confirmed by FE in [15] and will be used for all the optimizations in this paper.
C. Analytical optimization method From the torque expression (1), it can be seen that Lstk, Fs and
Λr are three variables directly related to output torque. Hence, five structural parameters, i.e., split ratio, stator and rotor pole arcs, stator/rotor yoke thicknesses, need to be optimized due to their close relationships with the aforementioned three variables, as shown in Fig. 4. Since the core saturation is neglected for analytical method, the stator/rotor yoke thicknesses are kept minimum according to the mechanical requirement and will be optimized later with the FE method.
With the design constraints listed in Table II, the optimal split ratio and stator/rotor pole arc ratios can be obtained with parametric calculation using the analytical model, as shown in Fig. 5. Finally, the globally optimized split ratio, stator and rotor pole arc ratios by analytical and FE methods are compared in Table III. Good agreements are found between analytical and FE results, indicating the feasibility of using analytical method for machine design under linear condition.
Further, it is worth noticing that the optimal rotor pole arc ratio is found to be 0.44 for all VFRMs under linear case. This is mainly due to the fact that the average torque of VFRMs is proportional to the fundamental rotor permeance component Λr1 (see equation (7)), and Λr1 peaks when the rotor pole arc ratio is 0.44, as shown in Fig. 4.
III. INFLUENCE OF MAGNETIC SATURATION In this section, the magnetic saturation is taken into account
by using the nonlinear FE method. Several 6s/4r and 6s/5r VFRMs are designed under different copper loss levels with pure global optimization method. The obtained optimal structural parameters are compared to those calculated by analytical method under linear case to investigate the influence of core saturation.
A. Split ratio The influence of load and magnetic saturation on the optimal
split ratio λ is shown in Fig. 7. It can be seen that the optimal split ratio shows an upward trend with the increase of copper loss. This can be explained by Fig. 8. On one hand, the increase of λ will lead to a reduction in slot area As and electric loading when the copper loss is fixed. On the other hand, owing to the reduced current, the magnetic saturation will be alleviated. Also, the rotor outer radius Rro becomes longer. A compromise should be made between these two aspects and the split ratio tends to increase under larger load condition for the sake of
alleviating the magnetic saturation. Compared with the linear case, the optimal split ratio will be increased by 1~1.2 times depending on the load condition.
(a)
(b)
Fig. 7. Variation of optimal split ratio against copper loss. (a) Optimal value. (b) Per unit value
Fig. 8. Relationship between split ratio and output torque.
B. Stator pole arc ratio The influence of load and magnetic saturation on the optimal
stator pole arc ratio βs is illustrated in Fig. 9. It is noted that the optimal βs is almost independent of load
condition. As shown in Fig. 10, an increase of βs will lead to smaller slot area and larger tooth width Wt, both of which will alleviate the stator saturation. However, a larger βs also means an increase in average airgap permenace Λs and more severe magnetic saturation. Hence, βs has insignificant influence on saturation. Its optimal value is almost constant with the increase of copper loss. In this case, the optimal value of βs for linear case is also applicable in nonlinear case.
Fig. 9. Variation of optimal stator pole arc ratio against copper loss.
156 CES TRANSACTIONS ON ELECTRICAL MACHINES AND SYSTEMS, VOL. 2, NO. 1, MARCH 2018
Fig. 10. Relationship between stator pole arc ratio and core saturation.
C. Rotor pole arc ratio The influence of load and magnetic saturation on the optimal
rotor pole arc ratio βr is illustrated in Fig. 11. The optimal value shows a downward trend with the increase of copper loss. Its variation range is between 0.33~0.44.
Fig. 11. Variation of optimal rotor pole arc ratio against copper loss.
Fig. 12. Relationship between rotor pole arc ratio and output torque.
(a)
(b)
Fig. 13. On-load rotor radial permenace distributions and variations of dc and 1st harmonics against rotor pole arc ratio for 6s/4r VFRM (Pcu=30W). (a) Rotor permeance distributions. (b) Variations of dc and 1st components.
This is mainly due to the influence of βr on the rotor permeance distributions, as shown in Fig. 12. By using the on-load permeance calculation method developed in [22], the rotor permeance distributions of different rotor pole arc ratios and the variations of the dc and 1st components with rotor pole
arc ratios are obtained in Figs. 13(a) and (b), respectively. On one hand, the average airgap permeance Λr0 drops with the decrease of βr. Hence, the magnetic saturation is alleviated. On the other hand, the decrease of βr will lead to a drop in Λr1, as well as the average torque. Considering both aspects, the optimal βr will drop from 0.44 to 0.33. Moreover, the more severe the saturation is, the smaller the optimal rotor pole arc ratio will be.
D. Stator/rotor yoke thickness Compared with the linear condition, the stator and rotor yoke thicknesses are expected to increase when the cores are saturated. The optimal yoke thicknesses can be obtained by global optimization method, as shown in Fig. 14.
(a)
(b)
Fig. 14. Variation of optimal stator and rotor yoke thicknesses against copper loss. (a) Stator yoke thickness. (b) Rotor yoke thickness.
IV. FAST OPTIMIZATION METHOD Based on the revealed influence of magnetic saturation on
the optimal structural parameters, a fast optimization method is developed by combining analytical and FE methods.
A. Conventional optimization method For conventional method, the torque calculation relies fully
on FE method. The global optimization module of ANSYS Maxwell 15.0 can be used to get the optimal specification. The procedure is:
Step 1: The parametric optimization is firstly used to get the rough variation range of all the structural parameters.
Step 2: All five structural variables are further globally optimized using the genetic algorithm.
In this case, 2~3 days are usually required.
B. Proposed fast optimization method The procedure of proposed fast optimization method is: Step 1: Set the rotor pole arc ratio as 0.44 (optimal value
HUANG et al: FAST DESIGN METHOD OF VARIABLE FLUX RELUCTANCE MACHINES 157
obtained from the analytical model) and stator/rotor yoke thicknesses as minimum values for mechanical consideration, the stator pole arc ratio and split ratio are optimized using analytical method.
Step 2: Globally optimize split ratio (1~1.2 times of linear optimal value), rotor pole arc ratio (0.33~0.44) and stator/rotor yoke thicknesses, whereas the stator pole arc ratio is fixed as the obtained optimal value of Step 1.
By using the fast optimization method, only 6~12 hours are required to obtain the optimal results, which are much shorter than the conventional method. The low time consumption is mainly benefited from the fast calculation speed of analytical method, less variable counts and narrowed variation ranges, as shown in Table IV.
TABLE IV COMPARISON OF CONVENTIONAL AND PROPOSED METHODS
Step Conventional method Proposed method
I
Parametric optimization Split ratio Stator pole arc ratio Rotor pole arc ratio (3 variables, 1~2 hours)
Analytical optimization Split ratio Stator pole arc ratio Rotor pole arc ratio (fixed to 0.44) (2 variables, 1~2 mins)
II
Global optimization Split ratio Stator pole arc ratio Rotor pole arc ratio Stator yoke thickness Rotor yoke thickness (5 variables, 2~3 days)
Global optimization Split ratio (1~1.2 times of step I) Stator pole arc ratio (identical to step I) Rotor pole arc ratio (0.33~0.44) Stator yoke thickness Rotor yoke thickness (4 variables, 6~12 hours)
C. Optimization of VFRMs with different stator/rotor pole number combinations Further, the developed fast optimization method is applied to
the optimization of the 6-stator-slot VFRMs with 2~20 rotor pole numbers according to the constraints listed in Table I. Their torque capabilities are shown in Fig. 15. Good agreement is found between the results of conventional and proposed methods while the proposed method can obtain the optimal design in much shorter time. Moreover, the 6s/7r and 6s/11r VFRMs are found to have the highest torque density. Considering the fact that 6s/7r VFRM has a much lower electrical frequency than 6s/11r VFRM under the same rotating speed, 6s/7r is the preferred stator/rotor pole combination under investigated specification.
Fig. 15. Variation of the average torque against rotor pole numbers for 6-stator-slot VFRMs with different rotor pole numbers (Pcu=30W).
V. EXPERIMENTAL VERIFICATION For experimental verification, a 6s/7r VFRM is prototyped,
as shown in Fig. 16. Its main specification is listed in Table V.
(a) Prototype
(b) Test rig
Fig. 16. Prototype and test rig of 6s/7r VFRM. (a) Prototype. (b) Test rig.
TABLE V MAIN SPECIFICATIONS OF PROTOTYPE 6S/7R VFRM
Parameter Value Parameter Value Number of phases 3 Stator outer diameter 90mm DC-bus voltage 48V Airgap length 0.5mm
Rated speed 400rpm Split ratio 0.56 Rated power 70W Stator pole arc ratio 0.31 Rated torque 0.82Nm Rotor pole arc ratio 0.366 Stack length 25mm Turns per coil (AC/DC) 144/144
It can be seen that both the armature and field windings are
concentrated types and wounded on all the stator teeth. During the experiment, the field winding is excited by a DC supply while the armature winding is connected to an inverter.
When the machine is operating at open-circuit and only the field windings are excited, the phase back-EMF can be measured, as shown in Fig. 17. Two different field currents are tested and good agreement can be found between FEA and experimental results. Although some harmonics can be observed from the back-EMF waveforms, they are close to sinusoidal for the 6s/7r VFRM.
Regarding the torque performance, the static torque is measured when both armature and field windings are excited by DC current with the following relationship: Idc=Ia=-2Ib=-2Ic. Since the armature and field windings share the same slot area and turns number, the RMS current of armature winding is equal to that of field winding to achieve the maximum torque. Again, the measurements match well with the FEA results, as can be seen from Fig. 18.
Then, the average torque of the prototype is also measured under different load condition, as presented in Fig. 19. It can be found that good agreement is found between FEA and measured results under low copper loss condition. When the copper loss is around rated condition, the measured average torque is slightly smaller than the FEA prediction. This is mainly due to the flux leakage of end windings.
Finally, the torque ripple performance of the prototype is measured when copper loss is 20W, as presented in Fig. 20. The torque profile is not fluctuating in ideal periodic way. This is mainly due to the measurement error and disturbance from the
0
0.2
0.4
0.6
0.8
1
1.2
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ave
rage
torq
ue (N
m)
Rotor pole number
Conventional method
Proposed method
Load machine Torque transducer VFRM
158 CES TRANSACTIONS ON ELECTRICAL MACHINES AND SYSTEMS, VOL. 2, NO. 1, MARCH 2018
load machine. Nevertheless, the peak-to-peak values of measurement and FEA prediction match with each other, verifying the feature of small torque ripple in 6s/7r VFRM [17].
Fig. 17. FEA predicted and measured phase back-EMFs at 400rpm for 6s/7r VFRM.
Fig. 18. FEA predicted and measured static torques when (Idc=Ia=-2Ib=-2Ic) for 6s/7r VFRM.
Fig. 19. FEA predicted and measured average torques under different copper loss for 6s/7r VFRM.
Fig. 20. FEA predicted and measured torque ripples for 6s/7r VFRM (Copper loss=20W).
VI. CONCLUSION In this paper, a fast optimization method is developed for
VFRMs by a combination of analytical and FEA methods. The capability and accuracy of analytical method in torque estimation under linear situation is firstly confirmed. The optimal rotor pole arc of linear case is found to be 0.44 for all the VFRMs. Then, the influence of saturation effect on optimal structural parameters is revealed. It is found that, depending on the load condition, (1) the split ratio will increase by 1~1.2 times of the linear optimal value; (2) the rotor pole arc ratio will decrease and vary within 0.33~0.44; (3) the stator pole arc will remain the same as its linear optimal value. Based on this, a fast optimization method is developed. Its capability is verified on the 6-stator-slot VFRMs having different rotor poles. Finally, a 6s/7r VFRM is prototyped and tested to verify the analyses.
REFERENCES
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-2
-1
0
1
2
0 60 120 180 240 300 360
Phas
e ba
ck-E
MF
(V)
Rotor position (Elec. deg.)
FEA
Measurement
Idc = 2.6A
Idc= 1.3A
-1
-0.5
0
0.5
1
0 60 120 180 240 300 360
Stat
ic to
rque
(Nm
)
Rotor position (Elec. deg.)
FEA
Measurement
Idc = 1.3A
Idc = 2.6A
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30
Ave
rage
torq
ue (N
m)
Copper loss (W)
FEA
Measurement
0
0.1
0.2
0.3
0.4
0.5
0.6
0 60 120 180 240 300 360
Tor
que
(Nm
)
Rotor position (Elec. deg.)
FEA Measurement
HUANG et al: FAST DESIGN METHOD OF VARIABLE FLUX RELUCTANCE MACHINES 159
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L.R. Huang received the B.Eng. and M.Sc. degrees in electrical engineering from Zhejiang University, Hangzhou, China, in 2012 and 2015, respectively. Since 2015, he has been working toward the Ph.D. degree in the Department of Electronic and Electrical Engineering, University of Sheffield, U.K.
His major research interests include design and application of reluctance machines and permanent magnet machines.
J.H. Feng (S’06) received his B.S. and M.S. degrees in Electrical Machinery Control from Zhejiang University, China in 1986 and 1989, respectively, and Ph. D. degree in Control Theory and Control Engineering from Central South University, China in 2008. Since 1989, he has been with CRRC Zhuzhou Institute Co. Ltd., Zhuzhou, China,
where he is presently the Vice President and Chief Technology Officer. He has published a number of journal and conference proceedings papers. His research interests are modeling, control, and communication of electrical systems, rail networks and high-speed trains. He is also a Guest Professor in Southwest Jiaotong University, Tongji University and Central South University.
S.Y. Guo is a professorial senior engineer. She got graduated from Central South University in December 1981, and serves as the chief technical expert in CRRC Zhuzhou Institute Co., Ltd. in the field of R&D of the electric machine systems for railway locomotive and electrical vehicle applications.
J.X. Shi received the B. Eng. and M. Sc. degrees in electrical engineering from South China University of Technology, Guangzhou, China in 2010 and 2013, respectively. Since 2013, he has been with CRRC Zhuzhou Institute Co., Ltd. His major research interests include design and application of permanent magnet
machines for electrical vehicle applications.
W.Q. Chu (SM’16) received the B. Eng. and M. Sc. degrees in electrical engineering from Zhejiang University, Hangzhou, China in 2004 and Huazhong University of Science and Technology, Wuhan, China in 2007, respectively, and the Ph.D. degree in the electronic and electrical engineering from The University of Sheffield, UK, in
2013. From 2007 to 2009, he was with Delta Electronics (Shanghai) Co. Ltd. From 2012 to 2014, he was a postdoctoral research associate with The University of Sheffield. Currently, he is a principal design engineer with CRRC Zhuzhou Institute Co. Ltd. His major research interests include the design and analysis of novel machines for wind power and electrical vehicle applications.
Z.Q. Zhu (M’90–SM’00–F’09) received the B.Eng. and M.Sc. degrees in electrical and electronic engineering from Zhejiang University, Hangzhou, China, in 1982 and 1984, respectively, and the Ph.D. degree in electrical and electronic engineering from The University of Sheffield, Sheffield, U.K., in 1991.
Since 1988, he has been with The University of Sheffield, where he is currently a Professor with the Department of Electronic and Electrical Engineering, Head of the Electrical Machines and Drives Research Group, Royal Academy of Engineering/Siemens Research Chair, Academic Director of Sheffield Siemens Wind Power Research Centre, Director of Sheffield CRRC Electric Drives Technology Research Centre. His current major research interests include the design and control of permanent-magnet brushless machines and drives for applications ranging from automotive to renewable energy.
